Nonlinear sine-Gordon Equation Research Articles

Acoustic and soliton propagation using fully-discrete energy preserving partially implicit scheme in homogeneous and heterogeneous mediums

This study presents an energy preserving partially implicit scheme for the simulation of wave propagation in homogeneous and heterogeneous mediums. Despite its implicit nature, the developed scheme does not require any explicit numerical or analytical inversion of the coefficient matrix. Theoretical analysis and numerical experiments are performed to validate the energy preserving properties of the fully-discrete scheme. Convergence analysis is also performed to assess the rate of convergence of the developed scheme. The efficiency and accuracy of the developed scheme are validated by numerical solutions of wave propagation in layered heterogeneous mediums. Furthermore, simulations of soliton propagation following nonlinear sine-Gordon and Klein-Gordon equations in homogeneous and heterogeneous mediums are discussed. Numerical solutions are also compared with the results available in the literature. The present method accurately resolves the physical characteristics of the chosen problems, competing well with existing multi-stage time-integration methods. Moreover, it has significantly lower computational complexity than the four-stage, fourth-order Runge-Kutta-Nyström method.

Novel approaches for nonlinear Sine-Gordon equations using two efficient techniques

ABSTRACT In this work, we obtained a new functional matrix using Clique-polynomials of complete graphs K n with n vertices and considered a new approach to solving the Sine–Gordon (SG) equation. The clique polynomial method transforms this equation into a system of algebraic equations. The solution will be drawn with the help of Newton Raphson’s method. Also, we employed the q-homotopy analysis transform method (q-HATM), which is the proper collision of the Laplace transform and the q-homotopy analysis method (q-HAM). To witness the reliability and accuracy of the considered schemes, some illustrations of the SG equation and double SG equation are considered. Here, the SG equation is solved easily and elegantly without using discretization or transformation of the equation by using the q-HATM. Also, in q-HATM, the presence of homotopy and axillary parameters allows us to have a large convergence region. The 3D surfaces of acquired solutions are drawn effectively. The tables of error analysis demonstrate the success of these methods.

Spectral algorithm for two-dimensional fractional sine-Gordon and Klein–Gordon models

Solitons and waves with memory effects are two examples of nonlinear wave phenomena that can be studied mathematically using the fractional nonlinear sine-Gordon equation. It is a modification of the traditional sine-Gordon equation that takes memory effects and nonlocal interactions into account by adding fractional derivatives. A more complex explanation of particle dynamics and interactions within relativistic quantum mechanics is made possible by the fractional Klein–Gordon model, a theoretical framework that expands the standard equation to include fractional derivatives. The study uses shifted Legendre–Gauss–Lobatto and shifted Legendre–Gauss–Radau collocation techniques to solve numerically two-dimensional sine-Gordon and Klein–Gordon models. The study handles two-dimensional sine-Gordon and Klein–Gordon models by extending a collocation approach using basis functions. It suggests a collocation technique that uses a suggested basis to automatically satisfy the conditions. The suggested methods’ spectral accuracy and efficiency are validated by numerical results.

Extension of optimal auxiliary function method to nonlinear Sine Gordon differential equations

In this paper, the new semi-analytical method called optimal auxiliary function method has been extended for finding the approximate solution of the non-linear Sine Gordon and Burger equations. The parameters of the auxiliary function’s optimum values control convergence, and they were determined using the well-known collocation approach. The Modified Adomian Decomposition Technique, Variational Homotopy Perturbation Method, Variational Iteration Method, and the Adomian Decomposition Method are used to compare the results produced by the suggested technique. It has been demonstrated that the suggested approach is straightforward, quickly convergent, effective, and consistent. The proposed method can also be made more accurate when making higher-order approximations.

Exploring Physics-Informed Neural Networks for the Generalized Nonlinear Sine-Gordon Equation

The nonlinear sine-Gordon equation is a prevalent feature in numerous scientific and engineering problems. In this paper, we propose a machine learning-based approach, physics-informed neural networks (PINNs), to investigate and explore the solution of the generalized non-linear sine-Gordon equation, encompassing Dirichlet and Neumann boundary conditions. To incorporate physical information for the sine-Gordon equation, a multiobjective loss function has been defined consisting of the residual of governing partial differential equation (PDE), initial conditions, and various boundary conditions. Using multiple densely connected independent artificial neural networks (ANNs), called feedforward deep neural networks designed to handle partial differential equations, PINNs have been trained through automatic differentiation to minimize a loss function that incorporates the given PDE that governs the physical laws of phenomena. To illustrate the effectiveness, validity, and practical implications of our proposed approach, two computational examples from the nonlinear sine-Gordon are presented. We have developed a PINN algorithm and implemented it using Python software. Various experiments were conducted to determine an optimal neural architecture. The network training was employed by using the current state-of-the-art optimization methods in machine learning known as Adam and L-BFGS-B minimization techniques. Additionally, the solutions from the proposed method are compared with the established analytical solutions found in the literature. The findings show that the proposed method is a computational machine learning approach that is accurate and efficient for solving nonlinear sine-Gordon equations with a variety of boundary conditions as well as any complex nonlinear physical problems across multiple disciplines.

Numerical solution of one-dimensional nonlinear Sine–Gordon equation using LOOCV with exponential B-spline

Numerical solution of one-dimensional nonlinear Sine–Gordon equation using LOOCV with exponential B-spline

A second-order accurate and unconditionally energy stable numerical scheme for nonlinear sine-Gordon equation

In this work, a second-order finite difference method is proposed to solve a nonlinear sine-Gordon equation. The constructed implicit scheme is proved to be unconditionally energy stable. A linear iteration algorithm is used to solve this nonlinear numerical scheme, and we prove that this iteration algorithm is convergent with a negligible constraint for time step. By constructing a suitable high-precision numerical solution, and using the inverse inequality and refinement constraint Δt≤Ch, the error estimate in L∞(0,T;L∞) norm of the fully discrete scheme is obtained. Several numerical examples are given to confirm the sharpness of our theoretical analysis.

Cubic B-spline method for non-linear sine-Gordon equation

Cubic B-spline method for non-linear sine-Gordon equation

Consistent travelling waves solutions to the non-linear time fractional Klein–Gordon and Sine-Gordon equations through extended tanh-function approach

In this study, the extended tanh-function method has been used to find further general travelling wave solutions for space-time fractional nonlinear partial differential equations, namely, the time fractional nonlinear Sine-Gordon equation and Klein–Gordon equation. The mentioned equations are useful for explaining a series of experimental seismic data, modelling of strain waves, and structures correlated to fault dynamics and the subduction slab, as well as slow earthquakes, episodic tremor, slow slip events, and tremor migration patterns. This method is used to attain exact solutions to a variety of nonlinear partial differential equations that were spatially temporal fractional. To get analytical solutions of the travelling wave type of certain nonlinear evolution equations, a power series in tanh was first utilized as an ansatz. These nonlinear fractional partial differential equations (NLFPDEs) are solved on a variety of non-rectangular domains. By using the fractional complex transform and the properties of the confirmable derivative, the two equations are reduced to ordinary differential equations. The solutions are sketched in 3D, and contour patterns, including king type, single soliton, double solitons, multiple solitons, bell shape, and other sorts of solutions. The achieved solutions are very much effective for explaining the above-stated phenomena.

Double Sumudu Transform Iterative Method for One-Dimensional Nonlinear Coupled Sine-Gordon Equation

In this paper, the combined double Sumudu transform with iterative method is successfully implemented to obtain the approximate analytical solution of the one-dimensional coupled nonlinear sine-Gordon equation (NLSGE) subject to the appropriate initial and boundary conditions which cannot be solved by applying double Sumudu transform only. The solution of the nonlinear part of this equation was solved by a successive iterative method, the proposed technique has the advantage of producing an exact solution, and it is easily applied to the given problems analytically. Two test problems from mathematical physics were taken to show the liability, accuracy, convergence, and efficiency of the proposed method. Furthermore, the results indicate that the introduced method is promising for solving other types of systems of NLPDEs.

DNA kinks behavior in the potential pit-trap

<abstract> <p>For better understanding the role of dynamic factors in the DNA functioning, it is important to study the internal mobility of DNA and, in particular, the movement of nonlinear conformational distortions -kinks along the DNA chains. In this work, we study the behavior of the kinks in the pPF1 plasmid containing two genes of fluorescent proteins (EGFP and mCherry). To simulate the movement, two coupled nonlinear sine-Gordon equations that describe the angular oscillations of nitrogenous bases in the main and complementary chains and take into account the effects of dissipation and the action of a constant torsion field. To solve the equations, approximate methods such as the quasi-homogeneous approximation, the mean field method, and the block method, were used. The obtained solutions indicate that two types of kinks moving along the double strand can be formed in any part of the plasmid. The profiles of the potential fields in which these kinks are moving are calculated. The results of the calculations show that the lowest energy required for the kink formation, corresponds to the region located between the genes of green and red proteins (EGFP and mCherry). It is shown that it is in this region a pit trap is located for both kinks. Trajectories of the kinks in the pit-trap and nearby are constructed. It is shown that there are threshold values of the torsion field, upon reaching which the kinks behavior changes dramatically: there is a transition from cyclic motion inside the pit-trap to translational motion and exit from the potential pit-trap.</p> </abstract>

Semi-analytical and numerical computation of fractal-fractional sine-Gordon equation with non-singular kernels

<abstract><p>In this article, we study the nonlinear sine-Gordon equation (sGE) under Mittag-Leffler and exponential decay type kernels in a fractal fractional sense. The Laplace Adomian decomposition method (LADM) is applied to investigate the sGE under the above-mentioned operators. The convergence analysis is provided for the proposed method. The results are validated by considering numerical examples with different initial conditions for both kernels and confirm the competence of the proposed technique. It is revealed that the obtained series solutions of the model with fractal fractional operators converge to the exact solutions. The numerical results converge to the particular exact solutions, proving the significance of LADM for nonlinear systems under fractal fractional derivatives. The absolute error analysis between the exact and obtained series solutions with both operators is shown in the tabulated form. The physical interpretations of the attained results with different fractal and fractional parameters are discussed in detail.</p></abstract>

Solutions of Two-Dimensional Nonlinear Sine-Gordon Equation via Triple Laplace Transform Coupled with Iterative Method

This article presents triple Laplace transform coupled with iterative method to obtain the exact solution of two-dimensional nonlinear sine-Gordon equation (NLSGE) subject to the appropriate initial and boundary conditions. The noise term in this equation is vanished by successive iterative method. The proposed technique has the advantage of producing exact solution, and it is easily applied to the given problems analytically. Four test problems from mathematical physics are taken to show the accuracy, convergence, and the efficiency of the proposed method. Furthermore, the results indicate that the introduced method is promising for solving other type systems of NLPDEs.

The energy-preserving time high-order AVF compact finite difference scheme for nonlinear wave equations in two dimensions

In this paper, energy-preserving time high-order average vector field (AVF) compact finite difference scheme is proposed and analyzed for solving two-dimensional nonlinear wave equations including the nonlinear sine-Gordon equation and the nonlinear Klein-Gordon equation. We first present the corresponding Hamiltonian system to the two-dimensional nonlinear wave equations, and further apply the compact finite difference (CFD) operator and AVF method to develop an energy conservative high-order scheme in two dimensions. The Lp-norm boundedness of two-dimensional numerical solution is obtained from the energy conservation property, which plays an important role in the analysis of the scheme for the two-dimensional nonlinear wave equations in which the nonlinear term satisfies local Lipschitz continuity condition. We prove that the proposed scheme is energy conservative and uniquely solvable. Furthermore, optimal error estimate for the developed scheme is derived for the nonlinear sine-Gordon equation and the nonlinear Klein-Gordon equation in two dimensions. Numerical experiments are carried out to confirm the theoretical findings and to show the performance of the proposed method for simulating the propagation of nonlinear waves in layered media.

Crank-Nicolson-DQM based on cubic exponential B-splines for the approximation of nonlinear Sine-Gordon equation

In this paper, Crank-Nicolson differential quadrature method based on cubic exponential B-spline (CExpB-spline) functions is presented to approximate the 1D nonlinear hyperbolic Sine-Gordon equation (SGE). The time derivative is discretized by the usual forward difference scheme while the differential quadrature method (DQM) is used to integrate the spatial derivatives. The discretization of the problem gives systems of linear equations. Three numerical examples are chosen to investigate the efficiency and accuracy of the method. It is observed that the proposed method provides excellent results than the existing methods. The rate of convergence (ROC) of the present method is obtained numerically showing that the method is second-order accurate in space.

Analytical Solution of Two-Dimensional Sine-Gordon Equation

In this paper, the reduced differential transform method (RDTM) is successfully implemented for solving two-dimensional nonlinear sine-Gordon equations subject to appropriate initial conditions. Some lemmas which help us to solve the governing problem using the proposed method are proved. This scheme has the advantage of generating an analytical approximate solution or exact solution in a convergent power series form with conveniently determinable components. The method considers the use of the appropriate initial conditions and finds the solution without any discretization, transformation, or restrictive assumptions. The accuracy and efficiency of the proposed method are demonstrated by four of our test problems, and solution behavior of the test problems is presented using tables and graphs. Further, the numerical results are found to be in a good agreement with the exact solutions and the numerical solutions that are available in literature. We have showed the convergence of the proposed method. Also, the obtained results reveal that the introduced method is promising for solving other types of nonlinear partial differential equations (NLPDEs) in the fields of science and engineering.

Dissipation-preserving Fourier pseudo-spectral method for the space fractional nonlinear sine–Gordon equation with damping

In this paper, an efficient numerical scheme is presented for solving the space fractional nonlinear damped sine–Gordon equation with periodic boundary condition. To obtain the fully-discrete scheme, the modified Crank–Nicolson scheme is considered in temporal direction, and Fourier pseudo-spectral method is used to discretize the spatial variable. Then the dissipative properties and spectral-accuracy convergence of the proposed scheme in L∞ norm in one-dimensional (1D) space are derived. In order to effectively solve the nonlinear system, a linearized iteration based on the fast Fourier transform algorithm is constructed. The resulting algorithm is computationally efficient in long-time computations due to the fact that it does not involve matrix inversion. Extensive numerical comparisons of one- and two-dimensional (2D) cases are reported to verify the effectiveness of the proposed algorithm and the correctness of the theoretical analysis.

Numerical solution of two-dimensional stochastic time-fractional Sine–Gordon equation on non-rectangular domains using finite difference and meshfree methods

The nonlinear Sine-Gordon equation is one of the widely used partial differential equations that appears in various sciences and engineering. The main purpose of writing this article is providing an efficient numerical method for solving two-dimensional (2D) time-fractional stochastic Sine–Gordon equation on non-rectangular domains. In this method, radial basis functions (RBFs) and finite difference scheme are used to calculate the approximate solution of the mentioned problem. The complexity of solving this problem arises from its high dimension, irregular area, stochastic and fractional terms. Finite difference technique is applied to overcome on the problem dimension, whereas interpolation method based on RBFs is the best idea for solving problems defined in irregular domains. The stochastic Sine–Gordon equation is transformed into elliptic stochastic differential equations using the finite difference method and meshfree method based on RBFs are used to approximate the obtained stochastic differential equation. Some numerical examples are included to investigate the efficiency and accuracy of the presented method.

Illustration of the Relativity Principle Using a Double Breather

The purpose of this article is to give more clarity to some relativistic laws, primarily the principle of relativity.These laws have been illustrated using linear waves in several of our previous articles. In this work, solutions of the nonlinear sine-Gordon equation in the form of double breathers are used for the same purposes. The exact two-breather solution and the approximate one that is convenient for analysis are presented. Moving breathers are derived from standing breathers through the active Lorenz transformations. Oscillations in a moving breather become out of phase. For each system, the distance between the amplitude maxima is taken as a unit length, and the oscillation period is taken as a time interval unit. It is shown that the observer associated with a moving breather and an observer associated with a standing breather see the same picture; an observer cannot distinguish the state of movement of his breather from rest. Therefore, in a frame of reference based on a moving breather, its oscillations are considered in-phase. In this case, time is determined as if the clocks were synchronized according to the Einstein method.Space-time coordinates of the same event in different frames of reference turn out to be related by the passive Lorentz transformations.

A study of nonlinear systems arising in the physics of liquid crystals, using MLPG and DMLPG methods

The study of liquid crystals is one of the active areas of physics research. In this paper, the MLPG and direct MLPG (DMLPG) methods are used for the numerical study of the coupled nonlinear sine–Gordon equations in two dimensions arising in the modeling of some phenomena in liquid crystals and superconductors. To approximate numerical integrals in the local weak forms, the MLS and GMLS approximations are used in MLPG and DMLPG methods, respectively. The distribution of regular and scattered points on rectangular and irregular domains has been used to extract the numerical results. By comparing the numerical results, it can be seen that the DMLPG methods are faster, more accurate, and more efficient than the MLPG methods. These are because the GMLS approximation uses the basis polynomials instead of the complex shape functions of the MLS approximation.